The manufacturer of padlocks of the brand ``K'' wishes to print a logo for his company. This logo has the shape of a stylized capital letter K, inscribed in a square ABCD, with side length one unit of length. We place ourselves in the orthonormal coordinate system $(A; \overrightarrow{AB}, \overrightarrow{AD})$.

\textbf{Part B: study of Proposal B}

This proposal is characterized by the following two conditions:
\begin{itemize}
  \item the line with endpoints A and E is a portion of the graph of the function $f$ defined for all real $x \geqslant 0$ by: $f(x) = \ln(2x + 1)$;
  \item the line with endpoints B and G is a portion of the graph of the function $g$ defined for all real $x > 0$ by: $g(x) = k\left(\frac{1 - x}{x}\right)$, where $k$ is a positive real number to be determined.
\end{itemize}

\begin{enumerate}
  \item a) Determine the abscissa of point E.\\
b) Determine the value of the real number $k$, knowing that the abscissa of point G is equal to 0.5.

  \item a) Prove that the function $f$ has as a primitive the function $F$ defined for all real $x \geqslant 0$ by:
$$F(x) = (x + 0.5) \times \ln(2x + 1) - x.$$
b) Prove that $r = \frac{\mathrm{e}}{2} - 1$.

  \item Determine a primitive $G$ of the function $g$ on the interval $]0; +\infty[$.

  \item It is admitted that the previous results allow us to establish that
$s = [\ln(2)]^2 + \frac{\ln(2) - 1}{2}$.\\
Does Proposal B satisfy the conditions imposed by the manufacturer?
\end{enumerate}